哈尔滨工程大学船舶结构力学课程习题集答案(11)

第6章 能量法

6.1题

1）方法一 虚位移法

考虑b),c)所示单位载荷平衡系统， 分别给予a)示的虚变形 ：

M(x)EIdx??d?

外力虚功为 ?W??虚应变能为

?V=?1??i?? ?1??j??M(x)MEI01l0(x)dx

?1?? =?EI?1??EI?0?Ril0ilx?Mi??0R?x1?idx

??Rx?M??Rx?dx0ii

?l???EI??=??l??EI???Mi3Mj3?Mj?l?1??M?M..........b)?j??i6?3EI?2?Mi?l?1?M?M...........c)??i??j6?3EI?2?

?由虚功原理：?W=?V 得：

?1??i?l? ?????3EIj??1????2?1?2??Mi???? Mj?1????

2）方法二 虚力法（单位虚力法） ?　梁弯曲应力：???＝MM?x?Iy

??＝?E＝?x?EIy

41

Mx??Mi??Mi?Mlj?xxl

?M?x??1?(1?0)

?M?x?Iy

给Mi以虚变化?Mi?1 虚应力为 ????＝虚余功：?W?＝?i?1

虚余能：?V*＝?（真实应变）?（虚应力）d?

?????1EI1EIM?x??M?x?yydxdydz EII2???l0l0M?x??M?x?dx?Ay2dA

j??Mi??Mi?M?1?x/l?dx ?x/l????

Qi?1??M?Mj??i3EI?2?l

同理：给Mj以虚变化?Mj?1，??Mi?0?可得（将i换为j）

?j??Mi???Mj??3EI?2?l

3）方法三 矩阵法（柔度法） 设??????,?p?????j?M?x?I??i??Mi???Mi?力?p?，虚?????,?????p????

?Mj???Mj?yx??Mi???????c??p?

Ml??j? ??????y?xy??Mi??Mi?Mj?x/l???1??II?l?式中?c??y??x??x??1??,??????I??l??l??（不妨称为物理矩阵以便与刚度法中几何矩阵

?B?对应）

??Mi???c???p???c???

?Mj???1?1虚应力????实应变?????D??????D??C??p?

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虚余功 ?W*??????p????p???????i?Mi??j?M虚余能 ?V*?????????d??????????d?

??TTTTj?

????p??C??D??C??P?d????p??TT?1T??T?1CDCd???????????p? ???于虚力原理：?W*??V*考虑到虚力??p?的任意性。得： ?????p???C??D???A??? p??C?d?T?1式中 ?A????C??D??C?d?——柔度矩阵（以上推导具有普遍意义）

?T?1对本题：

x??1??y??y?x??l?1??????I?x?EI??l????l??2??x???1??l?x?1l????d???l?EI0?x?x???1???l???l???A????x?x??1????l?l??dx2??x??????l??

由

1?l/3?l/?6l????EI??l/6l/3?3EI????1?1/21?/2? ?1????A??p?展开得： ??i?l ?????j?3EI?1/?2?Mi??1???M? 1??j???1/26.2题

方法一 单位位移法

???uj?ui?/l ， ??E??E?uj?ui?/l

ui/l??1/l设 ?ui?1，则 ?????

Ti?1???El?uj?ui???1/l?d???EAl2?l0?uj?ui?dx?EAl?ui?uj?

同理，令?uj? 可得

Tj?1???l?uEj?ui??1/l?d??EAl?uj?ui?

?Ti?EA?1即：????Tlj??1???1??ui???? 可记为 1??uj??p???K????

ijij 43

?K?为刚度矩阵。

方法二 矩阵虚位移法 设?pij????TiTj??T ??i???j?u1iu??jT

? {?}??uj?ui?1l?ui?/l???1??1???l?uj??B??????j i?式中 ?B????11?——几何矩阵

? ?????D??????D??B???ij?

设虚位移

??ij?????uiT?uj?? ， 虚应变 ??????B????ij?

TijijijT外力虚功 ?W??pij???????????p?

T?TT虚应变能 ?V?????????d??????????d?

?T ???????????B??D??Bij?ijd?

????i?jT?TB???????D??ij?Bd???????

ij ????i?K???j??

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